النقاط الثابتة قي جبر كلين

Conference Paper

توفيق, فردوس . 2015

• رابط المنشور على الويب: http://icmcmst.alpha-publishing.net
• اسم المؤتمر: “Mathematical and Computational Modelling in Science and Technology
• تاريخ المؤتمر: الأربعاء, تموز (يوليو) 8, 2015
• المنظمة الراعية: Izmir University, Izmir-Turkey

مستخلص المنشور

Fixed points in kleene Algebra
Fardous Toufic, Fairouz Tchier Mathematics department,King Saud University, Saudi Arabia. E-mail: ❢t♦✉❢✐❝❅❦s✉✳❡❞✉✳s❛ Abstract Kleene algebra is an algebraic system for calculating with sequential composition, choice and finite iteration. It was first introduced by Kleene in 1956 and further developed by Conway in 1971. It has reappeared in many contexts in mathematics and computer science. Its classical application has been within the theory of formal languages, it is one of many equivalent approaches to the description of regular languages. The fixed points play an important role in mathematics and computer science specially in the while-loop semantics. In this paper, we will give some relevant results about the fixed points in Kleene algebra. Relation Algebras We will give a definition of relational algebra and we recall some basic facts about fixed points. Most of our definitions are taken from [1, 2]. Definition 1. An abstract homogeneous relational algebra ¡ R,∪,∩, ¯,◦, T ¢ consists of a nonempty set R, whose elements are called relations, such that a) (R,∪,∩, ¯) is a complete, atomic Boolean algebra with zero element O, universal element L, and order ⊆, b) (R,◦) is a semigroup with precisely one unit element I, c) Schröder equivalences are valid i.e. for all relations Q,R and S, we have QR ⊂ S ⇐⇒Q T S ⊂ R ⇐⇒ SRT ⊂ Q. d) Tarski rule is valid i.e. LRL = L for all R 6= O. We will define fixed points of some function f and give their properties. Definition 2. A fixed point of a function f : S −→ S is an element s ∈ S for which f (s) = s. Kleene algebra Kleene algebra (K A) is an algebraic structure that captures axiomatically the properties of a natural class of structures arising in logic and computer science. It is named for Stephen Cole Kleene (1909−1994), who among his many other achievements invented finite automate and regular expressions, structures of fundamental importance in computer science. Kleene algebra is the algebraic theory of these objects, although it has many other natural and useful interpretations. Keywords : Relational algebra, Kleene algebra, fixed points, matrices, reflexive transitive closure. 2010 Mathematics Subject Classification : 03C05; 03C50; 03C52; 03B30. ICMCMST’2015, August 02-07, 2015 Izmir University-Turkey 59 International Conference Mathematical andComputationalModeling in Science andTechnology ICMCM